From length-preserving pushouts of graphs to one-surjective pullbacks of graph algebras

Abstract

The unions of directed graphs are the simplest examples of pushouts of directed graphs. The conditions under which they contravariantly induce surjective gauge-equivariant pullbacks of graph ${\mathrm{C}}^{\ast }$-algebras have been well studied and vastly instantiated in noncommutative topology (e.g., quantum balls and spheres). Herein, we go beyond the unions of graphs to systematically determine optimal conditions for more general length-preserving pushouts of graphs under which they contravariantly induce graded pullbacks of path algebras, Leavitt path algebras, and graph ${\mathrm{C}}^{\ast }$-algebras. Our pullbacks are surjective only on one side, as dictated by natural examples and K-theory. The proposed new approach enlarges the scope of applications from admissible subgraphs (also called quotient graphs) to generalizations of unlabeled foldings of Stallings and collapsing the line graphs of graphs to initial graphs. Moreover, we introduce the concept of locally derived graphs, which substantially extends the paradigm of derived graphs (or skew products of graphs), and use the projection foldings from locally derived graphs to their base (or voltage) graphs to obtain one-surjective pullbacks of graph ${\mathrm{C}}^{\ast }$-algebras.